Geometric and approximation properties of some singular integrals in the unit disk
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The purpose of this paper is to prove several results in approximation by complex Picard, Poisson-Cauchy, and Gauss-Weierstrass singular integrals with Jackson-type rate, having the quality of preservation of some properties in geometric function theory, like the preservation of coefficients’ bounds, positive real part, bounded turn, starlikeness, and convexity. Also, some sufficient conditions for starlikeness and univalence of analytic functions are preserved. Copyright © 2006 G. A. Anastassiou and S. G. Gal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let us consider the open unit disk D = {z ∈ C; |z| < 1} and A(D) = { f : D → C; f is analytic on D, continuous on D, f (0) = 0, f (0) = 1}. Therefore, if f ∈ A(D), we have ∞ f (z) = z + k=2 ak zk , for all z ∈ D. For f ∈ A(D) and ξ ∈ R, ξ > 0, let us consider the complex singular integrals
ξ Qξ ( f )(z) = π
π −π
1 +∞ iu −|u|/ξ Pξ ( f )(z) = f ze e du, z ∈ D, 2ξ −∞ iu f ze ξ +∞ f ze−iu ∗ du, z ∈ D, Qξ ( f )(z) = du, u2 + ξ 2 π −∞ u2 + ξ 2 2ξ 3 +∞ f zeiu Rξ ( f )(z) = du, z ∈ D, π −∞ u2 + ξ 2 2 1
Wξ ( f )(z) =
πξ
1 Wξ∗ ( f )(z) = πξ
π
−π
+∞ −∞
f zeiu e−u /ξ du,
2
f ze−iu e−u /ξ du, 2
z ∈ D,
z ∈ D, z ∈ D. (1.1)
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 17231, Pages 1–19 DOI 10.1155/JIA/2006/17231
2
Geometric and approximation properties
Here Pξ ( f ) is said to be of Picard type, Qξ ( f ), Qξ∗ ( f ), and Rξ ( f ) are said to be of PoissonCauchy type, and Wξ ( f ) and Wξ∗ ( f ) are said to be of Gauss-Weierstrass type. In the very recent papers [3–5], classes of convolution complex polynomials were introduced and their approximation properties regarding rates, global smoothness preservation properties, and some geometric properties like the preservation of coefficients’ bounds, positivity of real part, bounded turn, starlikeness, convexity, and univalence were proved. The aim of this paper is to obtain similar properties for the above-defined complex singular integrals. 2. Complex Picard integrals In this section, we study the properties of Pξ ( f )(z). Firstly, we present the approximation properties. Theorem 2.1. Let f ∈ A(D) and ξ ∈ R, ξ > 0. Then (i) Pξ ( f )(z) is continuous on D, analytic on D, and Pξ ( f )(0) = 0; (ii) ω1 (Pξ ( f ); δ)D ≤ ω1 ( f ; δ)D , for all δ ≥ 0, where ω1 ( f ; δ)D = sup{| f (z1 ) − f (z2 )|; z1 ,z2 ∈ D, |z1 − z2 | ≤ δ }; (iii) |Pξ ( f )(z) − f (z)| ≤ Cω2 ( f ; ξ)∂D , for all z ∈ D, ξ > 0, where ω2 ( f ; ξ)∂D = sup f ei(x+u) − 2 f eiu + f ei(x−u) ; x ∈ R, |u| ≤ ξ .
(2.1)
Proof. (i) Let z0 ,zn ∈ D be with limn→∞ zn = z0 . We get Pξ ( f ) zn − Pξ ( f ) z0 ≤ 1
2ξ
≤
1 2ξ
1 = 2ξ
+∞ −∞
+∞ −∞
iu f zn e − f z0 eiu e−|u|/ξ du
+∞ −∞
ω1 f ; zn eiu − z0 eiu D e−|u|/ξ du ω1
(2.
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